The scenario you are descibing is when a Cherokee citizen moved away from tribal jurisdiction -- essentially relinguished their citizenship to become a US citizen. They had to deny or hide that they were Indian to become US citizens and have voting rights, protection under the law, property ownership, or even to live in a particular State. The descendants of these persons cannot now claim Cherokee citizenship anymore than a person of Scottish descent can claim Scottish citizenship.

While traditionally the eastern tribes were matrilineal societies, when the tribes nationalized in the early 1800s, the matrilineal tie was not required for tribal citizenship and, as in the US, there were "naturalized" citizens who were not "Indian by blood", as they say. Native Americans were not citizens of the US until the last century and therefore were denied legal rights and protection (this includes intermarried whites). The States passed laws prohibiting ownership of property by Indians (including whites) or any person with "Indian blood".

There were a significant number of white Cherokee citizens who had all the rights of citizenship as any other Cherokee -- they owned improvements on the land (Cherokee didn't "own" land but they owned the improvements and had rights to exclusive usage as long as they maintained it), they voted, they held political office, they were subject to Cherokee laws. Under the current Cherokee Constitution (and I believe most federally-recognized tribes), there are no longer "naturalized citizens". You have to be a direct descendant of a "Cherokee by Blood" on the "Dawes Roll". This could change if the Freedmen's decendants are allowed citizenship -- it would logically follow that the descendants of intermarried whites would become citizens too.